Diffusion anisotropy of water in biological tissues can be conventionally quantified with the diffusion tensor (DT) and related indices such as the fractional anisotropy. DT can describe the diffusion displacement probability using a Gaussian distribution function. One of the main applications of the DT is tracing the white matter pathways in the brain using local estimates of the fiber orientations. However, in regions with complex fiber configurations, DT likely fails to describe the full directional information of the diffusion process. Most notably, the DT is not able to resolve fiber crossing, which occurs in many brain regions. A more complete depiction of the water diffusion displacement is given by the probability density function (PDF). PDF can be approximated using q-space imaging techniques, but these techniques require diffusion measurements for a large range of diffusion weightings and diffusion directions. To overcome these limitations, an orientation distribution function (ODF) has been provided for diffusion displacement probability distribution.
Several approaches have been proposed to estimate the ODF. One such approach is the q-ball imaging (QBI), which is based on a Funk transform of high angular resolution diffusion imaging (HARDI) data. The QBI approach has been extended to explore the spherical harmonic basis functions and multiple wavevector fusion. For brain imaging, each of these techniques has several limitations including the need for high b values (i.e., 3000 s/mm2 or above) and a large number of encoding directions or the assumption of specific diffusion properties for the investigated fiber populations.
Diffusional kurtosis is a quantitative measure of the degree to which the diffusion displacement probability distribution deviates from a Gaussian form. Diffusional Kurtosis Imaging (DKI) is a magnetic resonance imaging (MRI) technique for measuring this quantity. However, conventional DKI methods require substantial time (approximately 1 hour or more) for post-processing of the acquired images which may be a significant disadvantage in clinical practice.
Further, diffusion of water molecules in biological tissues can be conventionally quantified via diffusion tensor imaging (DTI). (See, e.g., Basser P J, Mattiello J, LeBihan D., MR diffusion tensor spectroscopy and imaging, Biophysical Journal 66:259-267 (1994)). DTI can be a valuable tool for noninvasive characterization of tissue microstructural properties (See, e.g., Basser P J., Inferring microstructural features and the physiological state of tissues from diffusion-weighted images, NMR in Biomedicine 8:333-3442-3 (1995); and Basser P J, Pierpaoli C., Microstructural and physiological features of tissues elucidated by quantitative-diffusion-tensor MRI, Journal of Magnetic Resonance B 111:209-219 (1996)). The DTI model can use a Gaussian approximation to the probability distribution governing the random displacement of water molecules. In many biological tissues, however, the displacement probability distribution can deviate considerably from a Gaussian form. Diffusional kurtosis imaging (DKI) can be an extension of DTI which can enable the characterization of this deviation by estimating the kurtosis of the displacement distribution (see, e.g., Jensen J H, Helpern J A, Quantifying non-Gaussian water diffusion by means of pulsed-field-gradient MRI, In Proceedings of the International Society for Magnetic Resonance in Medicine Annual Meeting, Volume 11, p. 2154, Toronto, Canada (2003); Jensen J H, Helpern J A, Ramani A, Lu H, Kaczynski K, Diffusional kurtosis imaging: The quantification of non-Gaussian water diffusion by means of magnetic resonance imaging, Magnetic Resonance in Medicine 53:1432-1440 (2005); and Lu H, Jensen J H, Ramani A, Helpern J A, Three-dimensional characterization of non-gaussian water diffusion in humans using diffusion kurtosis imaging, NMR in Biomedicine 19:236-247 (2006)), in addition to the estimation of the standard DTI-derived parameters. DKI has shown promising results in studies of human brain aging (see, e.g., Falangola M F, Jensen J H, Babb J S, Hu C, Castellanos F X, Di Martino A, Ferris S H, Helpern J A, Age-related non-Gaussian diffusion patterns in the prefrontal brain, Journal of Magnetic Resonance Imaging 28:1345-1350 (2008)), detection of cerebral gliomas (see, e.g., Raab P, al. e, Diffusional kurtosis imaging of cerebral gliomas—analysis of microstructural differences, Radiology, in press (2010)), and rodent brain maturation (see, e.g., Cheung M M, Hui E S, Chan K C, Helpern J A, Qi L, Wu E X, Does diffusion kurtosis imaging lead to better neural tissue characterization? A rodent brain maturation study, Neurolmage 45:386-392 (2009)). Moreover, the additional information provided by DKI has been exploited to resolve intravoxel fiber crossings (see, e.g., Lazar M, Jensen J H, Xuan L, Helpern J A, Estimation of the orientation distribution function from diffusional kurtosis imaging, Magnetic Resonance in Medicine 60:774-781 (2008)), thus improving upon DTI-based fiber tracking methods, for example.
An exemplary DKI model can be parameterized by the diffusion and kurtosis tensors from which several rotationally-invariant scalar measures can be extracted. The most common tensor-derived measures can be axial, radial, and mean kurtoses (see, e.g., Jensen et al., Quantifying non-Gaussian water diffusion by means of pulsed-field-gradient MRI (2003), supra.; Jensen et al, Diffusional kurtosis imaging: The quantification of non-Gaussian water diffusion by means of magnetic resonance imaging (2005), supra.; Lu H, supra.; and Hui E S, Cheung M M, Qi L, Wu E X, Towards better MR characterization of neural tissues using directional diffusion kurtosis analysis, NeuroImage 42:122-134 (2008)); mean, axial, and radial diffusivity (see, e.g., Basser P J. Inferring microstructural features and the physiological state of tissues from diffusion-weighted images (1995), supra.; and Song S-K, Sun S-W, Ramsbottom M J, Chang C, Russell J, Cross A H, Dysmyelination revealed through MRI as increased radial (but unchanged axial) diffusion of water, NeuroImage 17(3):1429-1436 (2002)); and fractional anisotropy (FA) (see, e.g., Basser P J. Inferring microstructural features and the physiological state of tissues from diffusion-weighted images (1995), supra.).
The interpretability of these metrics can be influenced by the estimation accuracy of the tensors. Noise, motion, and imaging artifacts can introduce errors into the estimated tensors. Sufficiently large errors can cause the tensor estimates to be physically and/or biologically implausible. For instance, the directional diffusivities can become negative, that is, the diffusion tensor can become non-positive definite (NPD). A consequence of NPD diffusion tensor estimates can be that FA values, which should range between 0 and 1, can exceed 1, particularly in high FA regions of the brain such as corpus callosum. (See, e.g., Koay C G, Carew J D, Alexander A L, Basser P J, Meyerand M E, Investigation of anomalous estimates of tensor-derived quantities in diffusion tensor imaging, Magnetic Resonance in Medicine 55:930-936 (2006)). Inaccuracies in the estimated diffusion and kurtosis tensors can also drive the estimated directional kurtoses outside of an acceptable range. Empirical evidence in the brain as well as idealized multi-compartment diffusion models suggest that directional kurtoses can be typically positive and also do not exceed a certain level depending on tissue complexity. (See, e.g., Jensen et al, Diffusional kurtosis imaging: The quantification of non-Gaussian water diffusion by means of magnetic resonance imaging (2005), supra.) The maximum allowable kurtosis can also be influenced by the maximum b-value used in image acquisition. (Id.).
Exemplary diffusion and kurtosis tensors were estimated using unconstrained nonlinear least squares (UNLS). (See, e.g., Lu H et al., Three-dimensional characterization of non-gaussian water diffusion in humans using diffusion kurtosis imaging (2006), supra.) In a related work, higher-order diffusion tensors were estimated using unconstrained linear least squares (ULLS). (See, e.g., Liu C, Bammer R, Acar B, Moseley M E, Characterizing non-Gaussian diffusion by using generalized diffusion tensors, Magnetic Resonance in Medicine 51:924-937 (2004)). These unconstrained schemes do not guarantee acceptable tensor estimates. To address this drawback in the context of DTI, the Cholesky decomposition has been utilized to impose the non-negative definiteness constraint on the diffusion tensor (see, e.g., Koay C G, Carew J D, Alexander A L, Basser P J, Meyerand M E, Investigation of anomalous estimates of tensor-derived quantities in diffusion tensor imaging, Magnetic Resonance in Medicine 55:930-93613, 15 (2006); and Wang Z, Vemuri B C, Chen Y, Mareci T H, A Constrained variational principle for direct estimation and smoothing of the diffusion tensor field from complex DWI, IEEE Transactions on Medical Imaging 23:930-939 (2004)) using either the ULLS or UNLS algorithms. Moreover, a parameterization has been exemplary in a UNLS framework to guarantee a positive diffusivity function in a fourth-order tensor-only model of diffusion. (See, e.g., Barmpoutis A, Hwang M S, Howland D, Forder J R, Vemuri B C, Regularized positive-definite fourth order tensor field estimation from DW-MRI, NeuroImage 45:S153-S162 (2009)).
Thus, there is a likely need for a tractable formulation of the tensor estimation problem in DKI, wherein the constraints on acceptable diffusivity and kurtosis parameters can be conveniently imposed. The estimation problem can be cast as linear least squares (LS) subject to linear constraints. The exemplary constrained linear LS (CLLS) formulation yields a convex objective function that avoids the problem of local minima, and permits efficient solutions via convex quadratic programming or a fast heuristic algorithm. The constraints ensure non-negative diffusivity along the imaged gradient directions, as well as guaranteeing that the directional kurtoses can be within a physically and biologically plausible range. The two algorithms exemplary for solving the CLLS problem strike different tradeoffs between the speed and accuracy of the solution as well as algorithm flexibility. The quadratic programming-based (CLLS-QP) algorithm exactly satisfies the constraints and can also handle an arbitrary number of diffusion weightings and different gradient sets for each diffusion weighting, but it does so at a moderately high computational cost. On the contrary, the heuristic (CLLS-H) algorithm produces an approximation to the optimal solution, but it accomplishes this at almost no computational overhead compared to the ULLS.
Thus, it can be desirable to provide exemplary embodiments of method, system and computer accessible medium for providing real-time diffusional kurtosis imaging, and to estimate the orientation distribution function, which can reduce or avoid at least some of the problems encountered by the conventional techniques as set out above.